Instead of solving (REF ) exactly, we apply randomized iterative solvers introduced in [1]}, which are competitive or better than deterministic solvers in many cases [2]}. The core idea is the sketch-and-project step. In particular, we let
\(K_t = \begin{pmatrix}B_t & G_t^T\\G_t & {0}\end{pmatrix}, \quad {\bar{\nabla }}\mathcal {L}_t = \begin{pmatrix}{\bar{\nabla }}_{{x}}\mathcal {L}_t\\c_t\end{pmatrix}, \quad {\tilde{z}}_t =\begin{pmatrix}{\tilde{\Delta }}{x}_t\\{\tilde{\Delta }}{\lambda }_t\end{pmatrix},\) 
